Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?

TL;DR

This paper investigates the geometric loci of triangle centers in elliptic billiards, establishing conditions under which these loci are ellipses or algebraic curves using algebraic and geometric methods.

Contribution

It introduces two rigorous methods to determine if the locus of a triangle center in an elliptic billiard is an ellipse, and proves algebraicity for rational center functions.

Findings

Loci of certain triangle centers are ellipses or higher-degree algebraic curves.

Two methods for proving elliptical loci are proposed: computer algebra and algebro-geometric.

Rational functions of side lengths lead to algebraic loci.

Abstract

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic